Suppose I have a survival data with the variables time: follow up time, event: event indicator(1 or 0) with 1 as an event and 0 as censored, treatment: treatment group (0 or 1) and covariates X1, X2, X3 AND X4.
First, I fit a logistic regression model to obtain the propensity scores. The outcome of logistic regression model is treament and X1, X2, X3 AND X4 are the predictors and obtain the propensity scores ps for each observations. Then, using inverse probability weighting, the weights wt are obtained as
treatment/ps + (1-treatment)/(1-ps).
Now, I want to fit a Cox proportional hazards regression model with these weights as follows:
model1 <- coxph(Surv(time,event) ~ treatment + X1 + X2 + X3 + X4, weights=wt).
Is model1 same as $\lambda(t|treatment,X_1,X_2,X_3,X_4)=wt*\lambda_0(t)* e^{\gamma *treatment+\beta_1*X_1+\beta_2*X_2+\beta_3*X_3+\beta_4*X_4}$ where $\lambda_0(t)$ is the baseline hazard function? What is the interpretation of adding these weights? Am I fitting weighted Cox proportional hazards regression model?
Best Answer
No, using the weights gives you a weighted estimator rather than a weighted model. The model is still $$\lambda(t,z)=\lambda_0(t)e^{z\beta}$$ but instead of estimating it by maximising the log partial likelihood you estimate it by maximising a weighted log partial likelihood. The contribution of observation $i$ to the log partial likelihood is multiplied by the weight $w_i$.
The resulting estimates of $\beta$ and $\lambda$ are the same as if you had $w_i$ identical copies of observation $i$ (except that $w_i$ doesn't have to be an integer), though the standard errors are not the same as if you had multiple identical copies.